Theorem mulgnnp1 13910

Description: Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.)

Hypotheses

Ref	Expression
mulg1.b	|- B = ( Base ` G )
mulg1.m	|- .x. = (.g ` G )
mulgnnp1.p	|- .+ = ( +g ` G )

Assertion

Ref	Expression
mulgnnp1	|- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ X ) )

Proof of Theorem mulgnnp1

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> N e. NN )
2		nnuz 9937	. . . . 5 |- NN = ( ZZ>= ` 1 )
3	1, 2	eleqtrdi 2331	. . . 4 |- ( ( N e. NN /\ X e. B ) -> N e. ( ZZ>= ` 1 ) )
4		simplr 533	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> X e. B )
5		simpr 110	. . . . . 6 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> u e. ( ZZ>= ` 1 ) )
6	5, 2	eleqtrrdi 2332	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> u e. NN )
7		fvconst2g 5920	. . . . . . 7 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) = X )
8		simpl 109	. . . . . . 7 |- ( ( X e. B /\ u e. NN ) -> X e. B )
9	7, 8	eqeltrd 2315	. . . . . 6 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. B )
10	9	elexd 2835	. . . . 5 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. _V )
11	4, 6, 10	syl2anc 415	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { X } ) ` u ) e. _V )
12		simprl 535	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> u e. _V )
13		mulg1.b	. . . . . . . 8 |- B = ( Base ` G )
14	13	basmex 13390	. . . . . . 7 |- ( X e. B -> G e. _V )
15		mulgnnp1.p	. . . . . . . 8 |- .+ = ( +g ` G )
16		plusgslid 13443	. . . . . . . . 9 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
17	16	slotex 13357	. . . . . . . 8 |- ( G e. _V -> ( +g ` G ) e. _V )
18	15, 17	eqeltrid 2325	. . . . . . 7 |- ( G e. _V -> .+ e. _V )
19	14, 18	syl 14	. . . . . 6 |- ( X e. B -> .+ e. _V )
20	19	ad2antlr 493	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> .+ e. _V )
21		simprr 537	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> v e. _V )
22		ovexg 6109	. . . . 5 |- ( ( u e. _V /\ .+ e. _V /\ v e. _V ) -> ( u .+ v ) e. _V )
23	12, 20, 21, 22	syl3anc 1278	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> ( u .+ v ) e. _V )
24	3, 11, 23	seq3p1 10880	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ ( ( NN X. { X } ) ` ( N + 1 ) ) ) )
25		id 19	. . . . 5 |- ( X e. B -> X e. B )
26		peano2nn 9295	. . . . 5 |- ( N e. NN -> ( N + 1 ) e. NN )
27		fvconst2g 5920	. . . . 5 |- ( ( X e. B /\ ( N + 1 ) e. NN ) -> ( ( NN X. { X } ) ` ( N + 1 ) ) = X )
28	25, 26, 27	syl2anr 290	. . . 4 |- ( ( N e. NN /\ X e. B ) -> ( ( NN X. { X } ) ` ( N + 1 ) ) = X )
29	28	oveq2d 6091	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ ( ( NN X. { X } ) ` ( N + 1 ) ) ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ X ) )
30	24, 29	eqtrd 2271	. 2 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ X ) )
31		mulg1.m	. . . 4 |- .x. = (.g ` G )
32		eqid 2238	. . . 4 |- seq 1 ( .+ , ( NN X. { X } ) ) = seq 1 ( .+ , ( NN X. { X } ) )
33	13, 15, 31, 32	mulgnn 13906	. . 3 |- ( ( ( N + 1 ) e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) )
34	26, 33	sylan 283	. 2 |- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) )
35	13, 15, 31, 32	mulgnn 13906	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) )
36	35	oveq1d 6090	. 2 |- ( ( N e. NN /\ X e. B ) -> ( ( N .x. X ) .+ X ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ X ) )
37	30, 34, 36	3eqtr4d 2281	1 |- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3705   X. cxp 4767   ` cfv 5372  (class class class)co 6075   1c1 8170   + caddc 8172   NNcn 9283   ZZ>=cuz 9900   seqcseq 10862   Basecbs 13330   +g cplusg 13408  ._gcmg 13899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-minusg 13786  df-mulg 13900

This theorem is referenced by:  mulg2  13911  mulgnn0p1  13913  mulgnnass  13937  gzsumconst  14120